Since Ruelle's inequality [1]} (see also [2]} for the random case on Banach spaces) tells us that
\(h({\mu }^{\prime }|\mathbb {P})\le \int \sum _{i} m_i\lambda ^{+}_{i}d{\mu }^{\prime }\) 
for any \(\Phi \) -invariant probability measure \({\mu }^{\prime }\)  projection to \(\mathbb {P}\) .
Then, it suffices to show that if \(\mu \)  is a random SRB measure the reverse inequality holds.
We assume that \(\mu \)  is ergodic here.
Let \(\eta \)  be constructed in proposition REF .
Recall that \(\eta _0:=\lbrace \lbrace \omega \rbrace \times \mathcal {B}\rbrace _{\omega \in \Omega }\) .
Since \(\eta \)  is \(\Phi \) -decreasing and \(\eta _0\le \eta \) , we have
\(h(\mu |\mathbb {P},\eta )&=\lim _{n\rightarrow \infty } \frac{1}{n}H_{\mu }(\vee _{i=0}^{n}\Phi ^{-1}\eta | \eta _0)\\&=\frac{1}{n}H_{\mu }(\Phi ^{-n}\eta |\eta )=H_{\mu }(\Phi ^{-1}\eta |\eta ).\) 

It suffices to show that
\(H_{\mu }(\Phi ^{-1}\eta |\eta )=\int \log J^{u}(\omega ,x)d\mu (\omega ,x)=\sum m_i\lambda _{i}^{+},\) 
where \(J^{u}(\omega ,x)=\det (Df_{\omega }(x)|E^{u}(\omega ,x))\) .
Note that \(\log J^{u}\in L^1(\mu )\) , the second equality follows immediately from the Birkhoff ergodic theorem and Proposition REF .
By Proposition REF  (c), we let \(\nu \)  be the \(\sigma \) -finite measure defined by
\(\nu (K)=\int v_{(\omega ,x)}((\eta (\omega ,x) \cap K)_{\omega }) d\mu (\omega ,x).\) 
By assumption, we have \(\mu \ll \nu \) . Let \(\rho =\frac{d\mu }{d\nu }\) , then for \(\mu \) -almost every \((\omega ,x)\)  one has
\(\rho (\omega ,y)=\frac{d\mu _{\eta (\omega ,x)}}{d\nu _{(\omega ,x)}}(y) \quad \text{for} \ \nu _{(\omega ,x)}\text{-almost every}\ y\in \eta (\omega ,x)_{\omega }.\) 
Recall that the information function \(I(\Phi ^{-1}\eta |\eta )\)  is
\(I(\Phi ^{-1}\eta |\eta )(\omega ,x)=-\log \mu _{\eta (\omega ,x)}((\Phi ^{-1}\eta )(\omega ,x)_{\omega }).\) 
Let \(L(\omega ,x)=\rho (\omega ,x)(J^{u}(\omega ,x))^{-1} (\rho (\Phi (\omega ,x)))^{-1}\) . For every measurable set \(E\subset \mathcal {B}\) .
On the one hand, since \(\eta \)  is \(\Phi \) -decreasing, one has
\(\mu _{(\Phi ^{-1}\eta )(\omega ,x)}(E)&=\dfrac{\mu _{\eta (\omega ,x)}((\Phi ^{-1}\eta )(\omega ,x)_{\omega }\cap E) }{\mu _{\eta (\omega ,x)}((\Phi ^{-1}\eta )(\omega ,x)_{\omega })}\\&=e^{I(\Phi ^{-1}\eta |\eta )(\omega ,x)}\int _{((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E} \rho (\omega ,y) d\nu _{(\omega ,x)}(y).\) 

On the other hand, by \(\mu \)  is \(\Phi \) -invariant, we have for \(\mu \) -almost every \((\omega ,x)\)
\(\mu _{(\Phi ^{-1}\eta )(\omega ,x)}(E)&=\mu _{\eta (\Phi (\omega ,x))}(f_{\omega }E)\\&=\int _{f_{\omega }(((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E)} \rho (\theta (\omega ),z) d\nu _{\Phi (\omega ,x)}(z)\\&=\int _{f_{\omega }(((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E)} \rho (\theta (\omega ),z) J^{u}(\omega ,f^{-1}_{\theta \omega }(z)) d((f_{\omega })_{\ast }\nu _{(\omega ,x)})(z)\\&=\int _{((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E} \rho (\Phi (\omega ,y)) J^{u}(\omega ,y) d\nu _{(\omega ,x)}(y),\) 

where we use the change of variables formula for induced measures on finite-dimensional manifolds in the third equality
(recall \(f^{-1}_{\theta \omega }(y)\)  is the unique point such that \(f_{\omega }(f^{-1}_{\theta \omega }(y))=y\) ).
Consequently, by \(E\)  is arbitrary and \( I(\Phi ^{-1}\eta |\eta )(\omega ,y)= I(\Phi ^{-1}\eta |\eta )(\omega ,x)\)  for any
\(y\in ((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\) , we get that for \(\mu \) -almost every \((\omega ,x)\)
\(I(\Phi ^{-1}\eta |\eta )(\omega ,y)=-\log L(\omega ,y) \quad \text{for} \ \nu _{(\omega ,x)}\text{-almost every}\ y\in ((\Phi ^{-1}\eta )(\omega ,x))_{\omega }.\) 
Therefore, for \(\mu \) -almost every \((\omega ,x)\)  one has
\(I(\Phi ^{-1}\eta |\eta )(\omega ,x)=\log J^{u}(\omega ,x)+\log \dfrac{\rho (\Phi (\omega ,x))}{\rho (\omega ,x)}.\) 
Since \(\log J^{u}\in L^1(\mu )\)  and \(I(\Phi ^{-1}\eta |\eta )(\omega ,x)\ge 0\) , one has \(\log ^{-} \frac{\rho \circ \Phi }{\rho }\in L^1(\mu )\) . Therefore, by Lemma REF  one has \(\int \log \frac{\rho \circ \Phi }{\rho }d\mu =0\) .
So, we have
\(H_{\mu }(\Phi ^{-1}\eta |\eta )=\int \log J^{u}(\omega ,x)d\mu (\omega ,x).\) 
This completes the proof.
